It's like walking into a timewarp. I'm wandering York's cobbly, touristy Micklegate, and I pop into the Oxfam bookshop. The Politics shelf is stuffed with Marxism, Gramsci, Lenin: all the lefty classics, as if the 80s had never gone away. Churches, more churches, a pretty brick-built alley with clematis montana tumbling down the walls. The brass plate on the building across the road does not look like that of a 21st-century visionary organisation, but that's how it's been described to me, by people in the know.

York's Centre for Separated Families is the only organisation of its kind in Britain. Unlike other agencies that step in when a family splits up, the centre refuses, as its website puts it, "to mirror the behaviour of many separating parents by taking sides". It insists that both mother and father are of equal importance in their children's lives, and has developed services to support all parents as they struggle, in often agonising circumstances, to do their best by the kids. The analysis is radical and sophisticated - forget all the boring blah about Manolos, if an idea such as post-feminism was ever to be useful, it would be doing something like this. And it isn't just analysis either. The centre offers advice and help with debt, benefits, health, family holidays, anger management, domestic violence and how to conduct a businesslike phone conversation with the person one now hates more than anyone in the world.

On the day I visited, most of the staff were out on a picnic. "With the Together group - it gives young parents a chance to have time together as a family which, otherwise, they seldom get. They can't get benefits so the mother usually has to stay with her mum, and young fathers get shut out. But I think young parents are great - so enthusiastic and full of energy and ideas. I wish the government would see that, instead of making them a big problem all the time."

Karen Richardson will be 44 this year, and has lived and worked in York all her life. A separated parent herself, she started working for a tiny charity called the One-Parent Families Support and Information Network in 1990. At first, the organisation followed the orthodoxy, reflected throughout government agencies and the family courts, that when a couple splits, it divides without remainder into the person who does all the childcare (in 90% of cases, the mother) and the person who brings in the money (assumed to be the dad). But Richardson started to wonder. "We like the question 'Why?' in this organisation. We ask the question 'Why?' a lot."

Richardson was aware of research indicating that after a family splits, the children who fare best are those who remain close to their non-resident parent, usually the father. At the same time, she noticed that economically straitened though single mothers often are, their ex-partners could not be said to be living the life of Riley either. Fathers moved out to bedsits, or to sleep on friends' settees, and even if they had a place their kids might stay, couldn't afford to have them. Child benefit and tax credits automatically go to the person the law calls the "parent with care". "So the fathers get ousted, and away they go ... Or they come back with a rights-based perspective" - by which she seems to mean, they get angry, they get vengeful, they get dressed up in Spider Man costumes. Which doesn't help them communicate any better with their exes, or to be better parents to their kids.

In 1998, the organisation embarked on a four-year partnership with Oxfam UK's Poverty Programme, which funded a project for the 10% of fathers locally who had full-time care of their children. It discovered that they, too, were struggling. The fathers reported being treated as "second-best", or with rudeness and suspicion. "We even had one young father who went to claim income support, only to be told he might have 'borrowed' the baby to claim benefits." Family services tend to be staffed mainly by women, yet women seem to find it difficult to behave respectfully towards lone fathers: "They're ignored, patronised, treated as invisible." One young man recalled a group where staff would whisk his baby off for a nappy-change without even asking him first.

"We did an evaluation and we recognised that much of our work excluded fathers. So we changed." Men get uncomfortable dealing with women-dominated organisations, and don't want to pop into places for cosy chats and cups of tea. So they set up a room with a library and computers, so men could look things up on their own. This has the knock-on benefit that many of the centre's excellent materials are available online, and there's phone and email advice for people who don't live locally.

We conduct our interview in one of the centre's advice rooms, with a computer, a wall of leaflets and a box of toys in the corner. "Oh my God," Richardson suddenly says at one point, "What on earth is that?" The ceiling has a large tarantula affixed to it - rubber, not real. The centre currently has eight staff, two men, six women, all with what Richardson calls "separation issues" in their backgrounds - "No, it's not deliberate, but people come to work here because they have an interest," and "No, our views don't line up according to gender. We're all way past that."

Richardson's own family was, she says, "ordinary working-class" - her mother worked for Rowntree's, making KitKats, and her father was a precision engineer. Karen left school for Rowntree's at 16, continuing her education informally, through the women's movement. In 1982 she took a job at a feminist publishing co-op; she made flower arrangements from home when her daughter was a baby. In 1990, she started working for the one-parent families group, directing its formal name-change and shift in direction in 2005.

She was single herself when she gave birth to Hannah in 1987. "I knew what I was getting into, but I knew I wanted to have children young." Hannah's biological father went off with new age travellers - "You remember the Battle of the Beanfield?" - and for the next few years she brought Hannah up herself, in a shared house. "So she had lots of mums and dads in a way - she was really loved and prized. It gave her a lot of confidence, I think - she really internalised a feeling of being valued."

When Hannah was five, Richardson moved in with the man who became her stepfather - "a wonderful man," Richardson says, and very much the man Hannah thinks of as her dad. A few years in, though, Richardson and Stepdad realised they worked better as friends than as a couple, and two years after that, Richardson met Nick, her current partner, himself a separated parent of two, whom she is due to marry next month. All involved were aware of how important Hannah's relationship with her stepdad was to her, and how fragile it would be if he and her mother split. So they didn't. Between 2000 and 2004 Hannah went on living in the family home, with Richardson and Stepdad moving in and out around her - an arrangement sometimes called "nesting". Now Hannah is 18 and about to go to drama college ("Can you put in her stage-name, Hannah Mae? She'd love that, thanks") and, says her mother, "absolutely sound in the world ... My belief is that if you're honest and clear and true with your kids, you can help them develop a clear, firm place to hold on."

It's quite funny, I observe, that a woman who was a committed 80s feminist should find herself campaigning to end discrimination against men. "Yes, but the books I loved were the ones about real equality, true partnership ..." She mentions Shulamith Firestone's The Dialectic of Sex (1970) and Marge Piercy's 1976 novel, Woman on the Edge of Time, both of which envisage utopian futures in which gender, as we understand it, no longer exists. Firestone looked forward to an "androgynous culture" supported by "the full development of artificial reproduction". Piercy imagined babies raised collectively, and suckled, as often as not, by men.

Richardson's own proposals, though less obviously science fiction, are as radical. "You can't ever get rid of heartbreak, but you can learn to put it to one side. That's part of what we're trying to do here: skilling up the parents of the future, get them carrying a relationships toolkit." It's an image both wonderful and disconcerting: new generations marching into the future, carrying that "clear, firm place" on their backs.

And Richardson's work, it seems to me, challenges something even more fundamental. While much recent feminism has followed the general consumerist drift into self-centredness, self-obsession, even, Richardson and her colleagues have quietly taken it somewhere else. "This is something I'm just beginning to crystallise, but ... it's about interdependence, isn't it? It's about the ongoing interdependence of mothers and fathers and children, and of community, too." That "clear, firm place" is neither individual nor unitary. It's made of relationships, and memories, and in a new way, family ties.